Fourier series (and transforms) are probably my most favorite mathematical tool in physics It has the power to transform nasty or complicated operators into so

Fourier series (and transforms) are probably my most favorite mathematical tool in physics It has the power to transform nasty or complicated operators into something more manageable: derivatives turn into multiplication by the wavevector, linear differential equations turn into finding the roots of a polynomial, integration over all of space turn into an (infinite) sum over all wavevectors, convolutions turn into pointwise multiplications, ... Later you learn that Fourier series are just one example of an application of the linear algebra concept of decomposing a vector onto an orthogonal basis, with the vectors in this case being functions, but the original will always have a special place in my heart